A Lie group is a differentiable manifold that has the structure of a group and that satisfies the additional condition that the group operations of multiplication and inversion are continuous.

A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. This definition is related to the fifth of Hilbert's problems, which asks if the assumption of differentiability for functions defining a continuous transformation group can be avoided.

compact Lie group | continuous group | group | Lie algebra | Lie groupoid | Lie-type group | Lorentz group | nil geometry | orthogonal group | semisimple Lie group | smooth manifold | Sol geometry | tangent space | vector field

graduate school level

Marius Sophus Lie